Properties

Label 2-4018-1.1-c1-0-88
Degree $2$
Conductor $4018$
Sign $-1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 2.23·5-s + 6-s − 8-s − 2·9-s − 2.23·10-s + 4.47·11-s − 12-s − 6.47·13-s − 2.23·15-s + 16-s − 3·17-s + 2·18-s + 4·19-s + 2.23·20-s − 4.47·22-s − 0.472·23-s + 24-s + 6.47·26-s + 5·27-s + 6.23·29-s + 2.23·30-s − 1.76·31-s − 32-s − 4.47·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.999·5-s + 0.408·6-s − 0.353·8-s − 0.666·9-s − 0.707·10-s + 1.34·11-s − 0.288·12-s − 1.79·13-s − 0.577·15-s + 0.250·16-s − 0.727·17-s + 0.471·18-s + 0.917·19-s + 0.499·20-s − 0.953·22-s − 0.0984·23-s + 0.204·24-s + 1.26·26-s + 0.962·27-s + 1.15·29-s + 0.408·30-s − 0.316·31-s − 0.176·32-s − 0.778·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(32.0838\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + 4.94T + 47T^{2} \)
53 \( 1 + 6.70T + 53T^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 - 2.23T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 - 2.94T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.144266088889297967355070265649, −7.21701023445518339067007334277, −6.63605297134577361262915198317, −5.98061143500565850461054530073, −5.25760794332877520231020417073, −4.47619137220741030084960298606, −3.11077197980668492156709920257, −2.28815566204990683242665584370, −1.33183159011323951243990733491, 0, 1.33183159011323951243990733491, 2.28815566204990683242665584370, 3.11077197980668492156709920257, 4.47619137220741030084960298606, 5.25760794332877520231020417073, 5.98061143500565850461054530073, 6.63605297134577361262915198317, 7.21701023445518339067007334277, 8.144266088889297967355070265649

Graph of the $Z$-function along the critical line